
When \(j=2\) , there is no normal space and \( X^{\prime }_{2,N}(t)\)  makes no sense. It is easy to see that (see, for example, Azaïs and Wschebor [1]})
\(\int _{S_2} \mathbb {E} \left(\det \left(X^{\prime \prime }_2(t)\right)\mid X(t)=x,\, X^{\prime }_2(t)=0\right)dt = \sigma _2(S) (x^2-1).\) 

When \(j=0\) , \(X^{\prime }_{0,N}(t)\- = X^{\prime }(t)\)  and:
\(\mathbb {E} \left(\mathbb {I}_{X^{\prime }(t)\in \widehat{C}_{t,0}}\mid X(t)=x \right)=\frac{\mathcal {A}(\widehat{C}_{t,0})}{2\pi };\) 
where \(\mathcal {A}(\widehat{C}_{t,0})\)  is the angle of the cone that is equal to the discontinuity of the angle of the tangent at the irregular point \(t\) .

When \(j=1\) , we consider a point \(t\)  on an edge \(L\)  of the exterior boundary. Note that in this case, the support cone \(C_t\)  is just a half-plane, so the event \(\lbrace X^{\prime }_{1,N}(t)\in \widehat{C}_{t,1}\rbrace \)  can be viewed as \(\lbrace X^{\prime }_{1,N}(t) \ge 0\rbrace \) .
At the point \(t\) , the second derivative along the curve can be expressed as:
\(X^{\prime \prime }_1(t)=X^{\prime \prime }_T(t)+C(t)X^{\prime }_{1,N}(t),\) 
where \(X^{\prime \prime }_T\)  is the second derivative in the tangent direction and \(C(t)\)  is the signed curvature at \(t\) .
It is easy to check that the covariance function of the vector \((X^{\prime \prime }_T,X^{\prime }_{1,N},X, X_1^{\prime })\)  is:
\(\left( \begin{array}{cccc}\textnormal {Var}(X^{\prime \prime }_T) & 0&-1&0\\0&1&0&0\\-1&0&1&0\\0&0&0&1\\\end{array} \right).\) 
Therefore, for such an edge \(L\) ,
\(\mathbb {E} \left(X^{\prime \prime }_1(t)\mathbb {I}_{X^{\prime }_{1,N}(t)\in \widehat{C}_{t,1}}\mid X(t)=x,\, X^{\prime }_1(t)=0\right)= \mathbb {E} \left(\left(-x+C(t)X^{\prime }_{1,N}(t)\right)\mathbb {I}_{X^{\prime }_{1,N}(t)\in \widehat{C}_{t,1}} \right)= \frac{-x}{2}+\frac{C(t)}{\sqrt{2\pi }}\) 
and
\(-\int _{L}\mathbb {E} \left(X^{\prime \prime }_1(t)\mathbb {I}_{X^{\prime }_{1,N}(t)\in \widehat{C}_{t,1}}\mid X(t)=x,\, X_1^{\prime }(t)=0\right)\frac{\varphi (x)}{\sqrt{2\pi }}dt=\frac{\sigma _1(L)x}{2\sqrt{2\pi }}\varphi (x)-\frac{\varphi (x)}{2\pi }\int _{L}C(t)dt.\) 
The quantity \(-\int _{L}C(t)dt\)  can be viewed as the variation of the angle of the tangent from the beginning to the end of this edge.
Since we complete a whole turn in the positive orientation:
\(\sum _{\textnormal { irregular points of the ext. boundary}}\mathcal {A}(\widehat{C}_{t}) \quad +\sum _{\textnormal {edges of the ext. boundary}}-\int _{L_i}C(t)dt=2\pi .\) 
For a point \(t\)  on an edge \(L_i\)  of the interior boundary (holes), the interpretation of the second derivative changes into:
\(X^{\prime \prime }_1(t)=X^{\prime \prime }_T(t)-C(t)X^{\prime }_{1,N}(t).\) 
Therefore,
\(-\int _{L_i}\mathbb {E} \left(X^{\prime \prime }_1(t)\mathbb {I}_{X^{\prime }_{1,N}(t)\in \widehat{C}_{t,1}}\mid X(t)=x,\, X_1^{\prime }(t)=0\right)\frac{\varphi (x)}{\sqrt{2\pi }}dt=\frac{\sigma _1(L_i)x}{2\sqrt{2\pi }}\varphi (x)+\frac{\varphi (x)}{2\pi }\int _{L_i}C(t)dt.\) 

